A052471 Number of noncaterpillar trees on n nodes (A000055-A005418).

0, 0, 0, 0, 0, 0, 1, 3, 11, 34, 99, 279, 773, 2103, 5661, 15160, 40373, 107355, 285059, 757273, 2013177, 5361100, 14303274, 38250297, 102538714, 275597098, 742674804, 2006661720, 5436008057, 14763754746, 40196603110, 109703958381, 300091975184, 822705857129

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Caterpillar Graph

Crossrefs

Cf. A000055, A005418.

Programs

Maple

with(numtheory):
b:= proc(n) option remember; `if`(n<=1, n,
      (add(add(d*b(d), d=divisors(j))*b(n-j), j=1..n-1))/(n-1))
    end:
a:= n-> b(n) -(add(b(k) *b(n-k), k=0..n)-`if`(irem(n, 2)=0,
        b(n/2), 0))/2 -ceil(2^(n-4) + 2^(iquo(n-2, 2)-1)):
seq(a(n), n=1..40); # Alois P. Heinz, May 18 2013

Mathematica

b[n_] := b[n] = If[n <= 1, n, (Sum[Sum[d*b[d], {d, Divisors[j]}]*b[n - j], {j, 1, n-1}])/(n-1)]; a[n_] := b[n] - (Sum[b[k]*b[n-k], {k, 0, n}] - If[ Mod[n, 2] == 0, b[n/2], 0])/2 - Ceiling[2^(n-4) + 2^(Quotient[n-2, 2] - 1)]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Feb 19 2016, after Alois P. Heinz *)

Extensions

a(14) and up from Eric W. Weisstein, Jul 17 2004.